Theorem algrf 15286

Description: An algorithm is a step function F : S --> S on a state space S. An algorithm acts on an initial state A e. S by iteratively applying F to give A, ( F ` A ), ( F ` ( F ` A ) ) and so on. An algorithm is said to halt if a fixed point of F is reached after a finite number of iterations.

The algorithm iterator R : NN0 --> S "runs" the algorithm F so that ( R ` k ) is the state after k iterations of F on the initial state A.

Domain and codomain of the algorithm iterator R. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

Hypotheses

Ref	Expression
algrf.1	|- Z = ( ZZ>= ` M )
algrf.2	|- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) )
algrf.3	|- ( ph -> M e. ZZ )
algrf.4	|- ( ph -> A e. S )
algrf.5	|- ( ph -> F : S --> S )

Assertion

Ref	Expression
algrf	|- ( ph -> R : Z --> S )

Proof of Theorem algrf

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		algrf.1	. . 3 |- Z = ( ZZ>= ` M )
2		algrf.3	. . 3 |- ( ph -> M e. ZZ )
3		algrf.4	. . . . 5 |- ( ph -> A e. S )
4		fvconst2g 6467	. . . . 5 |- ( ( A e. S /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) = A )
5	3, 4	sylan 488	. . . 4 |- ( ( ph /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) = A )
6	3	adantr 481	. . . 4 |- ( ( ph /\ x e. Z ) -> A e. S )
7	5, 6	eqeltrd 2701	. . 3 |- ( ( ph /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) e. S )
8		vex 3203	. . . . 5 |- x e. _V
9		vex 3203	. . . . 5 |- y e. _V
10	8, 9	algrflem 7286	. . . 4 |- ( x ( F o. 1st ) y ) = ( F ` x )
11		algrf.5	. . . . 5 |- ( ph -> F : S --> S )
12		simpl 473	. . . . 5 |- ( ( x e. S /\ y e. S ) -> x e. S )
13		ffvelrn 6357	. . . . 5 |- ( ( F : S --> S /\ x e. S ) -> ( F ` x ) e. S )
14	11, 12, 13	syl2an 494	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( F ` x ) e. S )
15	10, 14	syl5eqel 2705	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
16	1, 2, 7, 15	seqf 12822	. 2 |- ( ph -> seq M ( ( F o. 1st ) , ( Z X. { A } ) ) : Z --> S )
17		algrf.2	. . 3 |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) )
18	17	feq1i 6036	. 2 |- ( R : Z --> S <-> seq M ( ( F o. 1st ) , ( Z X. { A } ) ) : Z --> S )
19	16, 18	sylibr 224	1 |- ( ph -> R : Z --> S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   ZZcz 11377   ZZ>=cuz 11687   seqcseq 12801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  alginv  15288  algcvg  15289  algcvga  15292  algfx  15293  eucalgcvga  15299  eucalg  15300  ovolicc2lem2  23286  ovolicc2lem3  23287  ovolicc2lem4  23288  bfplem1  33621  bfplem2  33622